General synthesis of complex analogue filters
C. Cuypers, N.Y. Voo, M. Teplechuk and J.I. Sewell
Abstract: General methods for the synthesis of complex band-pass and band-stop analogue filters are developed for implementation by active-RC, gm-C, switched-capacitor (SC) and switched- current (SI) techniques. These principles are extended to synthesis procedures for log-domain complex realisations. Circuits for filters and group delay equalisers are presented for both ladder and cascade-biquad realisations. These have been incorporated into the filter design software XFILTER. Various designs are compared, noise and sensitivity responses being shown.
1 Introduction Re H (s) + Complex (polyphase) filters are finding many applications Re X (s) Re today [1–7], especially in the communications area. They are _ Y (s) essentially integrated circuit filters and, in order to cover the Im different frequency ranges of application, it is necessary to H (s) encompass different implementation techniques, such as input output active-RC, gm-C, switched-capacitor (SC), switched-current Im (SI) and log-domain. It is also important to develop a H (s) general synthesis method without undue restrictions on Im Im order and filter characteristic. X (s) Y (s) A complex band-pass transfer function HC(s)isderived Re from a low-pass transfer function H(s) by the frequency H (s) + shift method using the substitution (s-s josh) giving HC(s) ¼ H(s josh), where osh is the shift frequency. In Fig. 1 Schematic representation of a complex analogue filter general the coefficients of HC(s)arecomplex[5], therefore Re Im HC(s) can be represented as HC(s) ¼ H (s)+jH (s), where both real HRe(s) and imaginary HIm(s) parts are real techniques have been extended to include this with both transfer functions. ladder and biquad implementations. General implementa- The input and the output signals are complex tion of all these procedures in the XFILTER software is Re Im available and provides a useful design tool. Comparison of X ðsÞ¼X ðsÞþjX ðsÞ designs is shown in numerous examples. Y ðsÞ¼Y ReðsÞþjY ImðsÞ 2 Ladder-derived continuous-time complex band- The relationships between the input and output signals are pass filters Y ReðsÞ¼H ReðsÞX ReðsÞ H ImðsÞX ImðsÞ The nodal equation of a passive ladder is described by Y ImðsÞ¼H ReðsÞX ImðsÞþH ImðsÞX ReðsÞ 1 and represented in Fig. 1. ðsC þ s C þ GÞV ¼ J ð1Þ The frequency shift method, which leads to two identical where C, C and G are admittance matrices of capacitors, low-pass filters connected by a cross-coupling network, has inverse inductors and resistors, respectively. V and J are been used successfully [6, 7]. This is now formalised for vectors representing the nodal voltages and input current ladder-derived filters in all the above realisation techniques. sources. The system is decomposed into two related first- Extension to the design of band-stop filters follows easily order systems, which can be implemented, directly by and essentially comprises two identical high-pass networks active-RC or gm-C circuits. The decomposition [8] is with cross-coupling elements. Cascade biquad designs are performed on the left matrix C or the right matrix C. also produced and have some value as practical realisations, For a left matrix decomposition C ¼ ClCr then mainly because of their simplicity. Equalisation of group 1 delay in these complex realisations is often required and the C W ¼ GV GV ð JÞ l s r IEE, 2005 1 CrV ¼ W IEE Proceedings online no. 20040816 s doi:10.1049/ip-cds:20040816 For a right matrix decomposition C ¼ ClCr then Paper first received 14th July 2003 and in revised form 11th May 2004 1 C. Cuypers, M. Teplechuk and J.I. Sewell are with the Department of C V ¼ ðC W þ GV JÞ Electronics and Electrical Engineering, University of Glasgow, Glasgow G12 l s l 8LT, UK 1 N.Y. Voo is with the Department of Electronics and Computer Science, I DW ¼ CrV University of Southampton, Southampton SO17 1BJ, UK s
IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 7 where W is an auxiliary vector of intermediate variables and Applying the shift frequency transformation gives ID is the identity matrix. 1 Using the frequency shift transformation (s-s josh), the CV ¼ ðJ GV gAW þ joshCVÞ left matrix decomposition becomes s 1 C W ¼ ðgAT V þ jo C WÞ 1 L s sh L C lW ¼ ðCV joshC lW s Which leads to gm-C complex band-pass filter implementa- joshGV þ joshJÞ GV þ J tions utilising two cross-coupled low-pass ladder realisa- 1 tions, though the single value of transconductance is C V ¼ ðW þ jo C VÞ r s sh r sacrificed in the cross-coupling terms. and the right matrix decomposition 3 Ladder-derived sampled-data complex band- 1 pass filters CV ¼ ðC W þ GV J jo CVÞ s l sh For switched networks (SC and SI), the frequency shifting 1 jy I W ¼ ðC V þ jo WÞ mechanism [4] is expressed as HC(z) ¼ Hlp(ze ), where D s r sh y ¼ osh T,withosh the shift frequency and T the sampling The complex transfer functions are available from two sets time. Equation (1), after bilinear transformation and some manipulation, gives of equations. Left matrix decompositions 1 For real transfer functions: A þ FB þ D V ¼ J00 c 1 C W Re ¼ ðCVRe þ o C W Im where l s sh l Im Im Re Re 2 T þ oshGV oshJ Þ GV þ J A ¼ C þ C þ G; B ¼ 2T G; D ¼ 2G and T 2 1 C VRe ¼ ðW Re o C VImÞ J00 ¼ð1 þ zÞJ: r s sh r z 1 1 For imaginary transfer functions C ¼ and F ¼ 1 z 1 1 z 1 Im 1 Im Re represent forward and backward Euler integrators, respec- C lW ¼ ðCV oshClW s tively [8]. Re Re Im Im oshGV þ oshJ Þ GV þ J A left matrix decomposition is
Im 1 Im Re AlW ¼ ðFB þ DÞV 2ð JÞ C rV ¼ ðW oshC rV Þ s 1 ArV ¼WW Al ð JÞ
Right matrix decompositions In general A ¼ LU, A ¼ UL, A ¼ IA, A ¼ AI are the various For real transfer functions: decompositions. Substituting LDI integrator operators into the above Re 1 Re Re Re Im equations CV ¼ ðClW þ GV J þ oshCV Þ s 1 A W BV DV 2 J Re 1 Re Im l ¼ 1 ð Þ I DW ¼ ðCrV oshW Þ 1 z s z 1 A V ¼ W A 1ð JÞ For imaginary transfer functions: r 1 z 1 l 1 Applying a frequency shift (z-ze jy) to obtain the complex CV Im ¼ ðC W Im þ GV Im JIm o CV ReÞ s l sh transfer functions gives 1 1 jy I W Im ¼ ðC VIm o W ReÞ AlW ¼AlWz e BV DV D s r sh þ DVz 1ejy 2ð JÞþ2ð JÞz 1ejy The complex band-pass filter realisation therefore consists 1 jy 1 jy 1 of two identical real low-pass filters cross-coupled via ArV ¼ArVz e þ Wz e Al ð JÞ resistances (terms in o ). A range of active-RC implemen- 1 1 jy sh þ Al ð JÞz e tations is available from a number of decompositions [8]. jy A number of gm-C ladder decompositions are aimed at Substituting e ¼ cosy+jsiny will lead to two low-pass producing equal transconductance values, one of these, the filters with cross-coupling and modified internal topologies, TC topological decomposition [9], is used to produce a which increase the complexity of circuit design. However, T D complex band-pass filter. With C ¼ ADA ,whereD is a for a sufficiently high clock rate (oclk/osh440), cosy 1 diagonal matrix of inverse inductance values and A is an then incidence matrix, the following pair of equations is AlW ¼ðFB DÞV 2ð JÞþj 2ð JÞC sin y equivalent to (1), (g is a scaling conductance and 2 1 þ jAlWC sin y þ jDVC sin y C L ¼ g D ): A V WC A 1 J jA 1 J C sin y 1 r ¼ l ð Þþ l ð Þ CV ¼ ðJ GV gAWÞ s þ jArVC sin y þ jWC sin y 1 C W ¼ ðgAT VÞ Complex transfer functions are described by two sets of L s equations:
8 IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 For real transfer functions Applying the frequency shift operator and approximating Re 1 Re cos yD1 leads to the system equations AlW ¼ ðFB Dð1 z ÞÞV 2ð JÞ 1 þ z 1 Im Im Im AI ¼ f½BlW þ DI þ J 2ð J ÞF sin y AlW F sin y DV F sin y 1 z 1
Re Re 1 Re 1 Im þ jJc sin y þ jAIc sin y ArV ¼ W C Al ð J Þ Al ð J ÞF sin y Im Im ArV F sin y W F sin y I DW ¼ cBrI þ jI DWc sin y þ jBrIc sin y For imaginary transfer functions: Which again yield two identical cross-coupled low-pass Im 1 Im networks. AlW ¼ ðFB Dð1 z ÞÞV 2ð JÞ The same theory applies equally to the design of complex Re Re Re analogue band-stop filters, with the result that two identical þ 2ð J ÞC sin y þ AlW C sin y þ DV C sin y high-pass networks with cross-connecting elements are Im Im 1 Im 1 Re required. It is also possible to derive complex biquad ArV ¼ W C Al ð J ÞþAl ð J ÞC sin y sections for all four techniques. These have been produced Re Re and hence cascade realisations for high order complex filters þ ArV C sin yC þ W C sin y can be utilised. Now a complex SC band-pass circuit can be obtained from two identical low-pass circuits with cross-couplings. 4 Complex filter designs A right matrix decomposition is The above complex filter design strategies have been AV ¼ FðBlW þ DVÞþJ implemented in XFILTER [11] and the software has been 1 used to produce a range of designs to illustrate the wide W ¼CBrV 2Bl J variety of realisations that are viable by various IC where Bl and Br are the factors of the right-hand matrix B. techniques. In general B is decomposed as LU, UL, IA and AI. One of the basic criteria in deciding the feasibility of Applying the frequency-shift transformation and some integrated realisations of analogue circuits is component manipulation gives two sets of equations, which yield the spread. Table 1 gives typical spread figures for active-RC complex transfer functions. realisations of complex band-pass elliptic filters of high For real transfer functions: order; these are all right-LU decompositions. The resistance Re Re Re Re AV ¼ FðBlW þ DV ÞþJ spread of the 26th order active-RC is well beyond the Im Im realistic range for IC implementation. J F sin y AV F sin y Tenth order complex elliptic band-pass filters will be used Re Re 1 Re 1 Im as examples in the following designs. Table 2 summarises W ¼BrV C 2Bl J 2Bl J F sin y Im Im the performances of active-RC tenth order complex band- BrV F sin y W F sin y pass and conventional direct band-pass (two normally For imaginary transfer functions required in quadrature signal applications) ladder-derived Im Im Im Im filters. More components are required to realise the direct AV ¼ FðBlW þ DV ÞþJ band-pass circuits, otherwise performance parameters are þ JReC sin y þ AV ReC sin y similar, though the complex case has slightly better noise Im Im 1 Im 1 Re behaviour. Of course the inherent advantage of the complex W ¼BrV C 2Bl J þ 2Bl J C sin y Re Re þ BrV C sin y þ W C sin y Which again yield complex SC band-pass realisations Table 1: Component spreads of active-RC complex elliptic utilising two cross-coupled low-pass circuits. band-pass filters For SI circuits (1) is modified by scaling the voltage Order of Capacitance Resistance vector by 1 O to give filter spread spread ðsC þ s 1C þ GÞI ¼ J 10 24 131 After bilinear transformation and some manipulation, the 14 44 252 left matrix decompositions follow exactly as the SC left decompositions and yield the same design equations. These 18 72 409 lead to two identical SI low-pass filters with cross-couplings. 26 145 906 The right decompositions for SI implementations are modified to overcome dynamic-range scaling problems and commence with the equations [10]: Table 2: Performance parameters for 10th order active-RC 1 þ z 1 right-LU complex and direct elliptic band-pass filters AI ¼ f½B W þ DI þ J l 1 z 1 Complex Direct band- I DW ¼cBrI band-pass pass Substituting the integrator operators gives No. of components 78 116 1 1 þ z 1 AI B W DI J No. of amps 20 38 ¼ 1 ½ l þ þ 1 1 z 1 z Cspread 24 10.3 z 1 R spread 131 79.5 1/2 I DW ¼ BrI Max pass-band noise nV/(Hz) 58.5 212.2 1 z 1
IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 9 Table 3: Comparison of complex and direct gm-C 10th synthesis is well developed [12] and the complex log-domain order elliptic band-pass filters filter synthesis method applies a linear transformation to the existing method. Complex Direct band- Rewriting (1) band-pass pass Z
CV_ þ C V þ GV ¼ UVIN No. of components 82 118 No. of gm 54 44 where U is a column vector of input connections. C spread 5.9 185.8 Applying the linear frequency shift (s-s jo )gives Z sh gm spread 9.8 1 CV_ C V GV jo CV jo CV UV 2 Max pass-band noise nV/sqrt(Hz)1/2 46 268 þ þ sh sh ¼ IN ð Þ Only the right matrix decomposition yields useful circuits in the log-domain [8],soC is factorised as
0 C ¼ CLCR ð3Þ After introducing an intermediate variable −20 Z
X ¼ CR V ð4Þ −40 equation (2) can be rewritten as −60 CV_ þ CLX þ GV joshCV joshCLCRV ¼ UV IN ð5Þ
magnitude, dB −80 Equations (4) and (5) describe the linear system, which can be rewritten as 8 −100 _ > CV ¼ UVIN CLX GV <> − 120 ½ joshC V joshCLCRV ð6Þ 0 5 10 15 20 > :> frequency, kHz I DX_ ¼ CRV Fig. 2 Response of 10th-order SC complex band-pass filter where ID is the identity matrix. The externally linear vectors of V and X can be exponentially mapped to nonlinear internal vectors [13] filteristheabilitytorejectimage frequencies in receiver using applications. Table 3 shows a similar pattern for gm-C 1 Yi=k _ Yi=k _ realisations. However the transconductance spread in the Yi ) e ¼ W Yi and WYi ¼ e Yi ð7Þ complex case will necessitate many different control loops k to define the various gm values, at the cost of extra circuitry. where k ¼ VT for bipolar and k ¼ nVT for CMOS weak- Figure 2 gives the frequency response of an SC inversion implementation. Applying (7) to (6) results in a implementation of the complex band-pass filter, using a system of equations 8 X clock frequency of 660 kHz. The constituent circuits are > Cii > eVi=kV_ ¼ U eVin =k G eXi=k G eVi=k shown in Figs. 3 and 4. The switches associated with each > k i i Lij ii cross-coupling capacitor may be dispensed with when <> j Vi=k Vi=k clocked with the same waveforms as the two low-pass joshCije joshGLij GRij e > ð8Þ circuits (as in this realisation). They are shown because > > X P > IDij i situations arise when it is necessary to utilise an alternative :> k _ Vj=k e Xi ¼ GRij e clocking scheme in order to reduce overall component k j spread. Figure 5 shows the sum sensitivity of complex and ^ Vin =k direct SC implementations. A switched-current implemen- Setting Vin ¼ e and rewriting in log-domain W variables (8) becomes tation of the same complex filter shows almost identical sum 8 P sensitivity performance. Figure 6 gives typical noise > _ ^ > CiiWVi ¼ UiVin GLij WXj GiiWVi performances of complex and direct SC implementations. < j jo C W jo G G G W Figure 7 shows a complex band-stop response for a > sh Pij Vi sh Lij Rij ii Vi ð9Þ > _ transconductance-C realisation. : IDij WXi ¼ GRij WVj Using XFILTER it is possible to examine many j alternative circuit solutions and compare relative perfor- After ‘real’ and ‘imaginary’ part extraction, (9) becomes mance parameters, such as component spreads, sensitivity 8 P Re Re Re Re > CiiW_ ¼ UiV^ GL W GiiW and noise behaviour. > Vi in ij Xj Vi > j > Im Im > þoshCijW þ oshGL GR GiiW 5 General complex log-domain ladder synthesis > PVi ij ij Vi > Re Re > ID W_ ¼ GR W <> ij Xi ij Vj The design of complex log-domain filters offers a rather j P exciting extension of boundaries in analogue signal proces- C W_ Im ¼ U V^ Im G W Im G W Im ð10Þ > ii Vi i in Lij Xj ii Vi sing. The low-voltage, wide-dynamic-range, high-frequency > j > nature of the log-domain approach embedded in the > o C W Re o G G G W Re > sh ij Vi sh Lij Rij ii Vi > complex filter common to many quadrature receiver > P architectures should provide an efficient topology for :> I W_ Im G W Im Dij Xi ¼ Rij Vj front-end RF receivers. Direct log-domain ladder-filter j
10 IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 o e o e o 22 e o 25 23 24 16 e o 21 o e 14 C e C o C e C Re C3 C6 V O
C1 5 7 9 −m C4 −m C7 −m C e 4 6 8 C C2 C5 C
10 12 − − C8 m C9 m
11 13
e e e e e o 27 15 18 19 20 17 3 e C1 oooooC10 C C11 C12 − Re 1 V in 2
o C 1
a
o e o e o 49 e o 52 50 51 43 e o 48 o e 41 C e C o C e C Im C3 C6 V O
32 C1 34 36 C4 C7 −m −m −m C e 31 33 35
C2 C5 C C
37 39 −m −m C8 C9 38 40
e e e e e o 54 42 45 46 47 44 30 e C1o C10 o C ooo C11 C12 −1 Im V in 29
28 o C
b Fig. 3 Low-pass right-LU circuit (real and imaginary) for 10th order complex SC band-pass filter a Real b Imaginary
The system (10) presents a general form of a complex log- 5.1 First-order complex log-domain domain ladder filter. This can be easily extended [12] to integrator block include zero-producing sections and, hence, more general An integrator is a necessary building block and in the log-domain ladder-derived filters. These equations represent frequency domain the transfer function of a first-order two identical low-pass filters with cross-coupling to produce complex coefficient system can be expressed as [6] a log-domain complex band-pass filter or two identical Y A high-pass filters with cross-coupling to produce an equiva- HsðÞ¼ ¼ ð11Þ lent band-stop filter. X 1 þ Bs josh
IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 11 C C 100 o1 x1 e1 e x1 e complex 32 4 5 1 1 31 direct
80 e1 o1 o1 o1
60 o1 Cx2 e1 e Cx2 e 34 4 7 1 1 31 40
e1 o1 o1 o1 sensitivity, dB/%
20 C C o x3 e e x3 32 1 1 6 5 1 e1 33 0 81013 e o 1 1 o1 o1 frequency, kHz
C o x4 e Fig. 5 Sum-sensitivities of SC 10th-order elliptic band-pass circuits 34 1 1 Cx4 6 7 e1 e1 33
e o 1 1 o o 1 1 12 direct complex 10
Cx5 e Cx5 e 36 o1 e1 6 9 1 1 33 8 ½
e1 o1 o1 o1 V/(Hz) 6
C C noise, 4 o x6 e e x6 e 34 1 1 8 7 1 1 35 2
e1 o1 o1 o1 0 0 5 10 15 20 Cx7 e Cx7 frequency, kHz o1 e1 9 1 e1 35 36 8 a 16 direct e o o1 o1 1 1 complex
C C 12 o x8 e e x8 38 1 1 10 11 1 e1 37 ½
e1 o1 o1 o1 8 V/(Hz)
C C noise, o x9 e e x9 e 40 1 1 12 13 1 1 39 4
e1 o1 o1 o1 0 0 5 10 15 20 Fig. 4 Cross-couplings for 10th-order complex SC band-pass filter frequency, kHz b Cxn ¼ Cn siny Fig. 6 Noise performance of 10th- and 14th-order SC complex and direct band-pass circuits In the time domain this becomes a 10th-order b 14th-order BY_ þ Y joshY ¼ AX ð12Þ Nonlinear mapping of (12) leads to The ‘imaginary’ part of the system is _ Im ^ Im Im Re B Y =k Y =k Y =k Y =k BWY ¼ AX WY þ oshWY ð16Þ YYe_ þ e joshe ¼ Ae ð13Þ k and this can be shown diagrammatically, Fig. 8, which or in log-domain W variables indicates two identical subsystems with cross connections _ ^ between the integrator blocks. Comparison of (15) and (16) BWY þ WY joshWY ¼ AX ð14Þ with (10) shows a similar structure and these complex log- The ‘real’ part of the system is domain integrators can be used in the realisation of general complex log-domain filters. A variety of complex integrator _ Re ^ Re Re Im BWY ¼ AX WY þ oshWY ð15Þ types can be utilised in synthesis. One design based on a well
12 IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 0 6 Complex log-domain cascade-biquadratic filters
The alternative approach to realising a transfer function of − 20 any order is by cascading second- and first-(if odd order) order blocks. A complex log-domain biquadratic section is −40 shown in Fig. 10 and a first-order complex log-domain section is given in Fig. 11. These are derived from standard log-domain sections [9]. The value of cross-connection −60 current is expressed as Iosh ¼ oshCVT where C is capacitor magnitude, dB value, VT is thermal voltage and ost is shifting frequency. −80 7 Complex group-delay equalisers
−100 Applying the frequency-shift technique to the ladder- 0246810derived all-pass method [14] again leads to cross-coupled frequency, kHz structures in all the realisation techniques discussed. A log- Fig. 7 Frequency response of 12th-order gm-C biquad complex domain version is shown in Fig. 12. Y(s)isasingly band-stop filter terminated ladder which can be synthesised by the log- domain technique. Cross-coupled all-pass biquadratic • Re ∧ Re Re X Re W W Re X Y Y Y log A B ∫ exp Icf
− − 1 Io Io Ic Re 1 3 2 Re I I in out SH 1 1 − sCVt sCVt Id ∧ • SH Im Im Im Im Im X X W W Y Y Y Io log A B ∫ exp 2
− Io −I −1 Ish Ish 4 Ish sh Icf Fig. 8 First-order log-domain complex system I −I I o o3 c3 Im 1 Im I in Iout FDEM 1 1 sCVt sCVt −I FDEOUT d I o Io Re Re VoutP VoutN I o2 FDINTCAP Re Re I V I VinN o4 inP o Io Id CCId log domain FDIDC I I Fig. 10 Complex log-domain biquadratic section o o Io Io FDEM
FDEOUT If
FDEOUT 1 I I o 1 sCVt c FDEM Re Re I I I I I I o o o o in out FDIDC −I I I d d CCd Io Io −I I Im FDINTCAP Im sh sh VinP VinN Im Im VoutP VoutN If I Io FDEOUT o FDEM 1 I I o1 sCVt c Fig. 9 Complex log-domain integrator Im Im Iin Iout
−I known fully differential integrator [12] isshowninFig.9. d This fully differential structure is incorporated into log domain XFILTER software to provide fully differential in-phase and quadrature signals throughout. Fig. 11 First-order complex log-domain section
IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 13 1 Re 1 Re I in I Re out I Re in I out 1 I O 1 sCVt − 2 − G − 2 Y(s) − Ish − G Ish
− − G 2
1 IO 1 sCVt − G
Im Im I in 1 Iout
Y(s) Fig. 14 Complex log-domain all-pass first-order section
− 2
0 complex Im direct I Im −20 in I out − 1 40 log exp log domain −60 −80
Fig. 12 Complex log-domain ladder all-pass equaliser magnitude, dB −100
−120 1 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Re I Re frequency, MHz I I in out I 1 Fig. 15 Tenth-order Chebyshev filter frequency responses O 1 −2 sCVt
Table 4: Comparison of complex and direct log-domain −I 10th order Chebyshev band-pass filters O 3 IO 2 − Ish 1 Complex Direct band-pass band-pass sCVt
− No. of Transistors 440 840 Ish Ish No. of Capacitors 20 40
1 C spread 2.02 1.3 sCVt Total I 379.9 mA1.642mA Ish −I I spread 36.9 12.1 O 3 IO 2 Max pass-band noise pA/sqrt(Hz)1/2 83 70
−G −2
1 I for log-domain direct and complex realisations are shown in O 1 sCVt Fig. 15. These were computed by Spectre using real Im I Re transistor models (typical BSim3v3 model for a targeted in I out 1 0.6 mm complementary bipolar process). They illustrate the inherent difference between direct and complex realisations, Fig. 13 Complex log-domain all-pass biquad due to the application of a low-pass/band-pass transforma- tion in the former and the linear frequency shift in the latter. Table 4 shows typical circuit parameters, and the overall sections can similarly be derived and typical log-domain low noise performance of log-domain realisations is notice- versions are shown in Figs. 13 and 14. able. The noise responses in Fig. 16 were computed with SpectreRF. The noise performance for the complex 8 Log-domaindesignexample realisation is higher, possibly due to the larger number of transistors in each signal path. This behaviour varies with The synthesis procedures for both ladder and biquad filter type. The group delay responses in Fig. 17 show the complex log-domain filters and equalisers have been effective equalisation of group delay on the complex filter implemented in XFILTER. The frequency responses of a using two identical cross-coupled sixth-order log-domain typical tenth order 0.5 dB ripple Chebyshev band-pass filter all-pass equaliser networks.
14 IEE Proc.-Circuits Devices Syst., Vol. 152, No. 1, February 2005 90 tions of all types is dominated by values of the cross- complex coupling elements, and the spread in the low-pass or high- 80 direct pass constituent filters is usually well conditioned. Many 70 solutions can be examined in the design process and an optimum realisation can be selected. In particular, complex 60
, Hz log-domain filters have many interesting properties and 50 further work is continuing to evaluate these and improve
pA/sqrt 40 practical designs.
noise 30 10 Acknowledgments 20 Financial support from the Engineering and Physical 10 Sciences Research Council (GR/M 45252) is gratefully 0 acknowledged. 0.6 0.8 1.0 1.2 1.4 frequency, MHz 11 References Fig. 16 Tenth-order Chebyshev filter noise responses 1 Snelgrove, W.M., and Sedra, A.S.: ‘State-space synthesis of complex analog filters’. Proc. European Conf. on Circuit Theory and Design, ECCTD, The Hague, The Netherlands, 1981, pp. 420–424 2 Lui, Q., Snelgrove, W.M., and Sedra, A.S.: ‘Switched-capacitor 12th-order complex combined implementation of complex filters’. Proc IEEE Int. Symp. on Circuits 30 and Systems, ISCAS, San Jose, CA, USA, May 1986, pp. 1121–1123 3 Muto, C., and Kambayashi, N.: ‘A leapfrog synthesis of complex 12th-order complex equaliser analog filters’, IEICE Trans., 1993, E76-A, (2), pp. 210–215 25 4 Muto, C., and Kambayashi, N.: ‘A realisation of real filters using complex resonators’, Electron. Commun. Jpn., 3, Random Electron. Sci., 1993, 36,(3) 20 5 Zhang, X., Iwahashi, M., and Kambayashi, N.: ‘A novel narrow-band
s band-pass filter and its application to SSB communication’, IEICE µ Trans., 1997, E80-C, (7), pp. 1010–1015 15 6 Crols, J., and Steyaert, M.S.J.: ‘Low-IF topologies for high- performance analog front ends of fully integrated receivers’, IEEE 10th-order complex filter Trans. Circuits Syst. II, Analog Digit. Signal Process., 1998, 45, (3), pp. 10 269–282 group delay, 7 Minnis,B.J.,Moore,P.A.,Payne,A.W.,Davie,A.J.,andGreer, N.P.J.: ‘A low-IF, polyphase receiver for DECT’. Proc. IEEE Int. 5 Symp. on Circuits and Systems ISCAS, Geneva, Switzerland, May 2000, pp. 83–86 8 Ping, L., Henderson, R.K., and Sewell, J.I.: ‘A methodology for 0 integrated ladder filter design’, IEEE Trans. Circuits Syst., 1991, 38, 0.5 1.0 1.5 (8), pp. 853–868 frequency, MHz 9 Greer, N.P.J., Henderson, R.K., Ping, L., and Sewell, J.I.: ‘Matrix methods for the design of transconductor ladder filters’, IEE Proc., Fig. 17 Simulated complex group-delay responses Circuits Devices Syst., 1994, 141, (2), pp. 89–100 10 Ng, A.E.J., and Sewell, J.I.: ‘Feasible designs for high order switched- current filters’, IEE Proc., Circuits Devices Syst., 1998, 146,(5),pp. 297–305 9 Conclusions 11 Teplechuk, M.A. and Sewell, J.I. ‘XFILT reference manual v.4.0’, Univ. of Glasgow, Dept of Electronics and Elect. Eng., UK, 2004 12 Ng, A.E.J., Sewell, J.I., Drakakis, E.M., Payne, A.J., and Toumazou, Methods for the synthesis of complex analogue filters as C.: ‘A unified matrix method for systematic synthesis of log-domain both ladder and cascade-biquad realisations have been ladder filters’. Proc. IEEE Int. Symp. on Circuits and Systems, ISCAS 2001, Sydney, Australia, 2001, Vol. 1, pp. 149–152 described. These general routines have been implemented in 13 Frey, D.R.: ‘Log-domain filtering: an approach to current-mode the XFILTER software. They have been used to design a filtering’, IEE Proc. G, Circuits Devices Syst., 1993, 140, pp. 406–416 variety of complex band-pass and band-stop filters. Typical 14 Ng, A.E.J., and Sewell, J.I.: ‘Log-domain allpass group-delay equaliser design with XFILTER’. Proc. IEEE Int. Conf. on Electronics, Circuits examples are shown and performance results are presented. and Systems, ICECS, Dubrovnik, Croatia, Sept. 2002, Vol. 1, The spread in component values of complex filter realisa- pp. 125–128
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